Direct observation of hot-electron-enhanced thermoelectric effects in silicon nanodevices

The study of thermoelectric behaviors in miniatured transistors is of fundamental importance for developing bottom-level thermal management. Recent experimental progress in nanothermetry has enabled studies of the microscopic temperature profiles of nanostructured metals, semiconductors, two-dimensional material, and molecular junctions. However, observations of thermoelectric (such as nonequilibrium Peltier and Thomson) effect in prevailing silicon (Si)—a critical step for on-chip refrigeration using Si itself—have not been addressed so far. Here, we carry out nanothermometric imaging of both electron temperature (Te) and lattice temperature (TL) of a Si nanoconstriction device and find obvious thermoelectric effect in the vicinity of the electron hotspots: When the electrical current passes through the nanoconstriction channel generating electron hotspots (with Te~1500 K being much higher than TL~320 K), prominent thermoelectric effect is directly visualized attributable to the extremely large electron temperature gradient (~1 K/nm). The quantitative measurement shows a distinctive third-power dependence of the observed thermoelectric on the electrical current, which is consistent with the theoretically predicted nonequilibrium thermoelectric effects. Our work suggests that the nonequilibrium hot carriers may be potentially utilized for enhancing the thermoelectric performance and therefore sheds new light on the nanoscale thermal management of post-Moore nanoelectronics.


Supplementary Figures
. Structure and current characteristic curve of test Nano-constriction device.

Supplementary References 1-28
Supplementary Note 1. Nano-constriction device fabrication Fig. S1(a) shows the schematic structure of the Si nano-constriction device fabricated by electron beam lithography (EBL) and inductive coupled plasma-reactive ion etching (ICP-RIE). The etching depth of about 115 nm is measured by mechanical profilometry, ensuring the conducted film outside of the channel has been totally etched. A narrow constricted conductive region with 400 nm width (as shown in Fig. S1(b)) is connected to the source and drain via the aluminum (~90 nm)-silicon ohmic contacts (as demonstrated in Fig. S1(c)). The cross-shaped gold stripes (~100 nm) are served in SNoiM measurements for guiding the center of the focal spot and the narrowest position.

Fig. S1
Structure and current characteristic curve of test Nano-constriction device: (a) Optical micrograph for sketching the structure of the Si device and (b) SEM image of the narrowest region (~400 nm width); (c) I-V curve of nano-constriction Si device, the measured drain-source current I (red line) exhibits a good linear relationship with bias voltage Vb (as indicated by the green linear-fitted line), which demonstrates the good Ohmic contact between P-doped Si channel and drain/source electrodes.

Supplementary Note 2. Calculation of carrier concentration
P-doped Si film is fabricated into a Hall bar structure (as shown in Fig. S2(a)) for determining the carrier concentration and electric conductivity of the device, both the parameters are essential for the numerical simulations. The width (W) of the Hall bar channel is 100 μm, and the distance between the longitudinal contacts (VR-and VR+) is 340 μm.
The electron's concentration N3D can be calculated by: Where I, VH, B, e and d are the current, Hall voltage, applied magnetic field (±300 mT), elementary electric charge and thickness of P-doped Si film (90 nm), respectively. It should be noted that, to eliminate the contribution to VH from the inevitable misalignment between the pair of Hall contacts (VH-and VH+), the values of VH here under both positive and negative magnetic field has been subtracting the background (transvers voltage under 0 T). Fig. S2(b) clearly show the linear relationship of VH with I, where the data measured under both positive (+300 mT; red dots) and negative (-300 mT; blue dots) magnetic field almost coincide with each other, demonstrating the contribution from the misalignment has been well eliminated. The average value of | H | in S1 is acquired by linear fitting (gray line) from the measured data. Therefore, the concentration N3D is calculated about 0.95×10 19 cm -3 . The electrical conductivity can be deduced from: Where ̅ is the average resistance between the longitudinal contacts (VR-and VR+), which is obtained from the I-VR data by linear fitting (as shown in Fig. S2(c)). As results, is calculated about 1.74×10 4 S/m, which is consistent with the previously reported value for this concentration (P-doped) 1,2 .

Supplementary Note 3. Detecting electron temperature by Scanning Noise Microscope
To mapping the local temperature of the electron, a kind of scattering Scanning Nearfield Optical Microscope(s-SNOM), which can be also called Scanning Noise Microscope (SNoiM) 3-6 is applied for electron temperature measurement. Moving electrons can generate fluctuating electromagnetic (EM) evanescent field which is supposed to be related to the average of electrons' kinetic energy. This evanescent field will propagate to the far-field with a nanoscale tungsten tip approaching the surface of the sample, then collected by the confocal optical system and detected by a sensor. The nearfield signal detected by SNoiM can be written as 7,8 : Where ρ(z, ω) is EM local density of states, z is the distance between tip and sample surface, ω is the angular frequency of the EM wave, ℏis the Dirac constant and k B is the Boltzmann constant. While the EM local density of states would be merely dependent by the material with invariant z and ω , the electron temperature turns out to be the only parameter producing influence to the nearfield signal u(z, ω) . All the spatial profiles of u(z, ω) can be converted to electron temperature by comparison with signal under zero external electrical field (note that the experiment is completed under room temperature, so the electron temperature should be 0 = 300 K under 0 V): can be calculated. However, the nearfield signal under 0 V (V 0 ) is extremely weak and cannot be measured by current modulation method, so the tip-height modulation is applied for extracting V 0 and the decay curve of nearfield signal under 0 V and 10 V are shown in Fig. S3(b). There is also a peak in the narrowest area, however the temperature of the electron is higher than the lattice's results shown in Fig. 3. The Scanning Thermal Microscope (SThM) is equipped with a lock-in amplifier, which allows directly observing the certain thermal signals from lattice system with high sensitivity of temperature and spatial resolution (better than 50 mK and 50 nm) 5,9 , such as Joule heating, TE cooling/heating, and the total lattice temperature (TL). In order to explore the current-polarity dependence of TL under nonequilibrium condition, AC square-waves bias voltage of 10 V (413 Hz) with appointed polarity is supplied to the device (as illustrated in Fig. S4(a)) while the SThM scanning the thermal profile of a fixed line at the constriction (scanning distance is 5 μm). The one-dimensional TL distributions for both directions of current obtained by the above method are shown in Fig. S4(b), in which the profiles clearly exhibit the current-polarity dependence: the profiles are shifted toward the direction of the carrier (electron; n-type silicon) traveling while the peak values are almost the same, leading to clear split between the profiles. To determine the narrowest position (the middle point) of constriction, we suppose that the deviated distance of profiles with opposite polarities are the same (but in the opposite direction), as a result, the middle position is defined by the crossover point of the profiles (x0=2.54 μm) that marked by the black dashed line in Fig. S4(b). Consequently, the deviation of TL relative to the middle position in Fig. 1(c) is well defined by this method. Owing to the large gradient of TL in the vicinity of constriction, the current-polarity dependence would lead to a considerable temperature difference ( L + (x) − L − (x)) between positive ( L + (x)) and negative ( L − (x)) biased conditions, which can be obtained by means of square-waves modulated voltage with ±10 V (Fig.  S4(c)). As shown in Fig. S4(d), the thermal profile of L + (x) − L − (x)exhibit a strong antisymmetric feature with the opposite signs on both sides of constriction center, the maximum value approach to 3 K indicating the appearance of the considerable nonequilibrium thermoelectric (TE) effect in TL. In principle, the net TE component is equal to 10,11 : 2 Accordingly, the TE cooling/heating as shown in Fig. 2 and Fig. 3 that acquired from the original data lock-in amplifier by the above method should be additionally multiplied by 1 2 ⁄ . As a comparison, Figs. S4(e) shows the 1-D Te profiles measured with SNoiM under unipolar pulsed bias 0/+10V (red) and 0/-10 V modulated at 5 Hz. Unlike the data in Fig. S4(b) (SThM), no peak shift can be identified in Fig. S4(e) between positive and negative bias. Consistently, the bipolar pulsed bias (+10V/-10V modulated at 5Hz) as shown in Fig. S4(f) gives no discernible signal like SThM measurement (Fig. S4(d)).

Supplementary Note 5. Linear TE Peltier effect in Chromium-Silicon heterostructure
The conventional Peltier effect was predicted to appear at the junction composed of dissimilar materials with different Seebeck coefficients, the Peltier cooling/heating power at the interface is determined according to: ̇P eltier = ab , where ab = ab = ( a − b ) is the differential Peltier coefficient of the junction, a and b denote the absolute Seebeck coefficient of the materials, hence ̇P eltier = ( a − b ) (S2) In most cases of conventional thermoelectric devices, the temperature variation can be neglected, correspondingly the temperature T in equation S2 can be treated as a constant, hence the Peltier signals depend linearly on the current. Despite the Peltier effects have been experimentally observed in the various types of the junction, such as metalgraphene 12,13 , metal-organic film 11 , and metal-semiconductor 14,15 , the quantitative analysis for linear dependence on the current of Peltier effect is still lacking. More importantly, to emphasize the reliability of Peltier signal detection, and eliminate the possible artefacts that may lead to a third-power dependence of current in the nonequilibrium thermoelectric data, a conventical thermoelectric device with a metalsemiconductor junction structure is fabricated. As sketched in Fig. S5(a), the device is composed of a P-doped silicon (n-type: ~1×10 19 /cm 3 ) segment with two Chromium (Cr) electrodes which are fabricated on silicon substrate (high-resistivity). The Peltier signals here are detected by using the AC square-waves bias voltage (as described in S4) with 213 Hz, and the Joule heating are measured by the sinusoidal-wave-modulated voltage (213 Hz), details for measuring method are described in the method. As shown in Fig. S5(d), when the bias voltage is supplied to the device, the Peltier cooling/heating effect appears at the junction as the current flow through the Cr/Si interfaces, owing to the dissimilar Seebeck coefficient of the P-doped (~1×10 19 /cm 3 ) Si (about hundreds of -μV/K) 2,16,17 and Cr (~20 μV/K) 18 according to the equation S2, where the sign of the Peltier signals are determined by the current direction and the sign of the differential Seebeck coefficient ( SiCr ). It should be noted that the Peltier signals are slightly concentrated toward the edges of Cr contacts, such phenomenon has also been observed in the metal-graphene heterojunction which is attributed to the current crowding induced non-uniform current density over a finite length in metal contacts 12 (as indicated in Fig. S5(a) right by red arrows). Whereas the Joule heating is concentrated around the center of the silicon segment (Fig. S5(c)) due to the relatively large resistivity of the doped silicon and the more concentrated current density (Joule-Lenz law). Fig. S5(e) and (f) show the 1D thermal profiles of Joule heating and Peltier cooling/heating under various bias voltages of 2-4 V that are measured along the junction, which allowed for quantitative analysis the signals. The peak values of Joule heating extracted from 1D profiles under various voltages are shown in Fig. S5(g), clearly showing quadratic dependence of the current (replace the voltage owing to the linear relationship between the bias voltage and current, as indicated in Fig. S5(b)) by the fitted line. In contrast to the nonequilibrium TE effect, the Peltier signals (absolute maximum values of Peltier cooling) here exhibit the linear relation with current (as indicated by linear fitting), consistent with the theoretical prediction from equation S2.

Supplementary Note 6. Simulation for nonequilibrium feature between electron and lattice
To explore the nonequilibrium signature between electron and lattice in nanoconstruction of Silicon under high electrical field, we adopted a recently developed simulation method 19 based on the two-temperature model for extracting the distributions of Te and TL from the respective subsystems. The simulations are performed by commercial Multiphysics software (COMSOL). Considering the practical computing time, the modeled geometry is simplified by a constriction structure (Fig. S7(a)) with the narrowest width of 400 nm (Fig. S7(b)), and the thickness of the substrate is down to 5 μm for the same reason. To ensure that the simulated distributions of electric field and current density around the narrowest region are in accord with those in the real device under the same bias conditions, we choose the current source instead of the voltage source for simulations, where the currents under various bias voltages are deduced from the I-V curve (see Fig. S1(c).
In this model, the conducting layer (P-doped Silicon film) is physically separated into electron-and the lattice layers, as shown in Fig. S7(c). The electron layer represents the electron subsystem, which can be characterized by the free electron density 3D , the electrical conductivity e , the electron thermal conductivity e , and the electronic specific heat Ce. 3D~1 × 10 19  time and e * = (3 2 ⁄ ) 2D B is the electronic specific heat per unit area). After released to the lattice subsystem, the heat transferred within the lattice by − L ∇ L (x, y) or spreads into the substrate via the thermal resistance ℎ I (we assume 0 Km 2 /W for ℎ I , owing the homoepitaxy of Si film) between lattice and substrate, and eventually spread to the heat sink (300 K) through the thermal conductivity of substrate Sub . Parameters used in the model are summarized in Table S1. Figure. S8 shows the simulation of the electric field strength | | at b = 10 V ( = 2.51 mA). The 2D distribution (Fig. S9(a)) of | | is intensively concentrated at the narrowest region with the maximal value approach of 25 kV/cm with a sharp 1D profiles ( Fig. S8(b)) along the channel, as indicated by the red dashed line in Fig. S8(a).
The nonequilibrium signatures between electrons and lattice at Vb=10 V are reproduced in the simulation results. As shown in Fig. S9, the distributions of e (Fig.  S9(a)) and L (Fig. S9(b)) extracted from the respective layers exhibit the similar features with the experiments (as shown in Fig. 1) where the peak values of e (~ 1600 K) and L (~340 K) are well matched with the experimental results (~1500 K and ~320 K). Moreover, the contour of the hot spot for L is broader than that for e , such a feature is clearly elucidated by plotting 1D profiles of Te and TL together in Fig. S9(c), reproducing the similar line feature in Fig. 1(c). The dissimilarity in line shapes between Te and TL is attributed to the differences in specific heat and thermal conduction between electron and lattice by orders of magnitude (see Table S1).    (b) Simulated lattice temperature profiles for +10 V and -10 V bias voltages. The deviation is dramatically smaller than our experimental data (Figs. 2d-h, Fig.3b).
(c) Derived Thomson signal profile (Vb=10 V) with the amplitude of the asymmetric curve being only roughly ~4 % of the real observed signal under the same bias (Fig. 3b).

Supplementary Note 8. Extracting the nonequilibrium thermoelectric signals by simulation
The simulations for the nonequilibrium Thermoelectric effects are implemented by a strategy that merges the electron and lattice layer as a whole uniform conducting layer (90-nm thick) where the lateral dimensions are consistent with that of the twotemperature model (as shown in Fig. S9(a)). This model includes heat transfer in solid, electrical current and thermoelectric effects. The main equations solved for this system is: Where , , and are the material properties of density, heat capacity, and electrical/thermal conductivities, respectively. , , , and are the temperature, heat density, Peltier coefficient, current density, and electric field.
The above equation applies for the quasi-equilibrium case where distinguishing e and L become unnecessary. In present work, SThM and SNoiM results show unambiguously the occurrence of nonequilibrium condition, i.e., e ≫ L . As a consequence, the thermal conductance term (∇ • ( ∇ )) in above equation should be renewed to include both electron and lattice contributions; the thermoelectric term (−∇ • ( )) should be replaced with nonequilibrium thermoelectric coefficient using electron temperature e because thermoelectric transport is realized by transport of hot electrons with effective temperature ( e ). Considering the steady-state which SNoiM mainly probes, the above equation can be transformed into,  S11), which are in agreement with the experimental data (as shown in Fig. S4(b)). Hence, the distributions of ∆ Joule and ∆ TE can be separately extracted by: ∆ Joule (x, y) = ∆ L + (x, y) + ∆ L − (x, y) 2 ∆ TE ± (x, y) = ∆ L ± (x, y) − ∆ L ∓ (x, y) 2 as shown in Fig. 3(d)-(f). Inaddition, it should noted that the simulated peak values of lattice temperature (~325 K, Vb=±10 V) by this model is comparable to that in the twotemperature model (~343 K, Vb=10 V), suggesting the reliability of these models.  The sample Vth signal is proportional to the tip temperature ( ∆ ) rise (i.e. temperature gradient) with negative polarity in region I (e) and positive in region III (f) according to equation S5. The temperature rise of the tip is calibrated and found to be proportional to the square of tip bias as shown in (g). In the nanoconstricted region II, no apparent sample Vth can be obtained for both without (h) and with +10 V (i) source-drain bias training and the absence of the sample Vth signal is also seen in the line profiles (j). Artificial signal appears at mesa edges due to symmetry-breaking in hot-tip-induced electron diffusion (red and blue in (h) and (i)).
To measure the local Seebeck coefficient distribution, an active heated-probe local thermovoltage measurement is carried out via the heated SThM tip (NP-SThM-02, TSPNANO). The probe is heated by an AC drive (5 MHz sine wave in our experiment) and temperature of tip apex is obtained by reading the DC thermovoltage ( tip,th ) signal of the probe. When the heated tip contact with the sample surface, the local temperature gradient ∇ induced by the tip would lead to the thermovoltage according to: We scan the hot probe along the metal/P-doped Si boundaries and the constriction of Pdoped Si channel, recording the thermovoltage of sample (according to Eq. S5). Change of Seebeck coefficient will cause thermovoltage under the temperature gradient. At the two boundaries of electrode and sample, opposite voltage is observed for positive and negative sides, as expected. Besides, by changing the temperature of the tip, liner response of thermovoltage with the temperature of the tip is shown (Figs. S12 (e), (f)), confirming the functionality of the measurements. However, no clear voltage signal has been observed at the nano-construction region for both before (Fig. S12(h)) and after ( Fig. S12(i)) driving the device with 10 V bias. We therefore conclude that there is no apparent change of Seebeck coefficient at the nanoconstruction.